Subgroups — #29566840

Order 15616.

306 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758185	C₄	1	2	yes	—
5	C₂×C₄×#92418	C₄	1	2	yes	—
6	C₄×#11758185	—	2	1	no	—
7	C₄×#11758185	—	2	1	no	—
8	C₈×#92418	—	2	1	no	—
9	C₈×#92418	—	2	1	no	—
10	C₂²×C₁₆×C₆₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₆₁	—	2	1	no	—
13	C₄×C₁₆×C₆₁	—	2	1	no	—
14	C₂²×#92418	C₈	1	4	yes	—
15	C₂×#11758185	—	2	2	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	C₈	1	4	yes	complemented
18	C₄×#92418	—	2	2	no	—
19	C₄×#92418	—	2	2	no	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	—	2	2	no	—
24	C₂×C₁₆×C₆₁	—	2	2	no	—
25	C₂×C₁₆×C₆₁	C₈	1	4	yes	complemented
26	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
39	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
40	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₆₁	M₁₆	1	2	yes	—
43	C₂×#92418	—	2	4	no	—
44	C₂×#92418	—	2	4	no	—
45	C₂×#92418	C₁₆	1	8	yes	complemented
46	C₁₆×C₆₁	—	2	4	no	—
47	#11758185	—	2	4	no	—
48	C₂³×C₆₁	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
54	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₆₁	M₃₂	1	4	yes	—
59	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
60	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
61	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
62	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
63	#92418	—	2	8	no	—
64	#7136	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	C₈²	1	1	yes	characteristic
66	C₂²×C₆₁	#322	1	1	yes	characteristic
67	C₂²×C₆₁	#336	1	1	yes	characteristic
68	C₂²×C₆₁	C₄×C₁₆	1	2	yes	—
69	C₂²×C₆₁	#346	1	2	yes	—
70	C₄×C₆₁	#322	1	1	yes	characteristic
71	C₄×C₆₁	#322	1	1	yes	characteristic
72	C₄×C₆₁	#336	1	1	yes	characteristic
73	C₄×C₆₁	#336	1	1	yes	characteristic
74	C₄×C₆₁	C₂²⋊C₁₆	1	4	yes	—
75	C₄×C₆₁	C₄⋊C₁₆	1	4	yes	—
76	C₂×C₄×C₁₆	—	61	1	no	—
77	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
78	C₂×C₈²	—	61	1	no	—
79	C₂×C₆₁	#1273	1	1	yes	characteristic
80	C₂×C₆₁	C₈×C₁₆	1	1	yes	characteristic
81	C₂×C₆₁	#1311	1	1	yes	characteristic
82	C₂×C₆₁	#1312	1	2	yes	—
83	C₂×C₆₁	#1374	1	2	yes	—
84	C₂²×C₁₆	—	61	2	no	—
85	C₂²×C₁₆	Dic₆₁	1	2	yes	—
86	C₂×C₄×C₈	—	61	2	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	122	1	no	—
91	C₄×C₁₆	—	122	1	no	—
92	C₈²	—	122	1	no	—
93	C₈²	—	122	1	no	—
94	C₆₁	#7136	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	61	4	no	—
96	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	61	4	no	—
102	C₂×C₁₆	—	122	2	no	—
103	C₂×C₁₆	—	122	2	no	—
104	C₂×C₁₆	#92418	1	4	yes	complemented
105	C₄×C₈	—	122	2	no	—
106	C₄×C₈	—	122	2	no	—
107	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
108	C₄×C₈	D₂₄₄	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
111	C₂²×C₄	C₈×D₆₁	1	2	yes	—
112	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
113	C₂×C₈	—	61	8	no	—
114	C₂×C₈	—	122	4	no	—
115	C₂×C₈	—	122	4	no	—
116	C₂×C₈	C₈×D₆₁	1	2	yes	—
117	C₂×C₈	#11758189	1	2	yes	—
118	C₂×C₈	C₂×#92418	1	2	yes	—
119	C₂×C₈	#11758194	1	2	yes	—
120	C₂×C₈	#11758196	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
122	C₂×C₈	#11758198	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
124	C₄²	#11758189	1	2	yes	—
125	C₄²	#11758196	1	1	yes	characteristic
126	C₄²	#11758198	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₁₆	—	122	4	no	—
129	C₂³	C₈×Dic₆₁	1	1	yes	characteristic
130	C₂×C₄	#14519684	1	1	yes	characteristic
131	C₂×C₄	C₈×Dic₆₁	1	1	yes	characteristic
132	C₂×C₄	C₄×#92418	1	1	yes	characteristic
133	C₂×C₄	#14519687	1	1	yes	characteristic
134	C₂×C₄	#14519687	1	1	yes	characteristic
135	C₂×C₄	#14519688	1	1	yes	characteristic
136	C₂×C₄	#14519689	1	2	yes	—
137	C₂×C₄	#14519706	1	2	yes	—
138	C₂×C₄	C₁₆×D₆₁	1	4	yes	—
139	C₂×C₄	#14519716	1	4	yes	—
140	C₈	—	122	8	no	—
141	C₈	#14519689	1	2	yes	—
142	C₈	C₂²⋊#92418	1	2	yes	—
143	C₈	#14519706	1	2	yes	—
144	C₈	C₄⋊#92418	1	2	yes	—
145	C₂²	C₈×#92418	1	1	yes	characteristic
146	C₂²	#16839484	1	1	yes	characteristic
147	C₂²	#16839516	1	1	yes	characteristic
148	C₂²	C₁₆×Dic₆₁	1	2	yes	—
149	C₂²	#16839537	1	2	yes	—
150	C₄	#16839483	1	1	yes	characteristic
151	C₄	#16839484	1	1	yes	characteristic
152	C₄	#16839516	1	1	yes	characteristic
153	C₄	#16839517	1	1	yes	characteristic
154	C₄	#16839541	1	4	yes	—
155	C₄	#16839580	1	4	yes	—
156	C₂	#21443646	1	1	yes	characteristic
157	C₂	C₁₆×#92418	1	1	yes	characteristic
158	C₂	#21443735	1	1	yes	characteristic
159	C₂	#21443738	1	2	yes	—
160	C₂	#21443896	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₆₁
commutator	C₂×C₆₁	C₈×C₁₆
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₄×C₈
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#7136
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
17. C₂×#11758185	C₈	semidirect
25. C₂×C₁₆×C₆₁	C₈	semidirect
45. C₂×#92418	C₁₆	semidirect
94. C₆₁	#7136	semidirect
104. C₂×C₁₆	#92418	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#11758185	2
C₂×C₈×#92418	2
C₂×C₄×C₁₆×C₆₁	2
#7136	61